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Theorem itg1addlem4 23686
Description: Lemma for itg1add . (Contributed by Mario Carneiro, 28-Jun-2014.)
Hypotheses
Ref Expression
i1fadd.1 (𝜑𝐹 ∈ dom ∫1)
i1fadd.2 (𝜑𝐺 ∈ dom ∫1)
itg1add.3 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))
itg1add.4 𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))
Assertion
Ref Expression
itg1addlem4 (𝜑 → (∫1‘(𝐹𝑓 + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
Distinct variable groups:   𝑖,𝑗,𝑦,𝑧   𝑦,𝐼   𝑦,𝑃,𝑧   𝑖,𝐹,𝑗,𝑦,𝑧   𝑖,𝐺,𝑗,𝑦,𝑧   𝜑,𝑖,𝑗,𝑦,𝑧
Allowed substitution hints:   𝑃(𝑖,𝑗)   𝐼(𝑧,𝑖,𝑗)

Proof of Theorem itg1addlem4
Dummy variables 𝑤 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 i1fadd.1 . . . . 5 (𝜑𝐹 ∈ dom ∫1)
2 i1fadd.2 . . . . 5 (𝜑𝐺 ∈ dom ∫1)
31, 2i1fadd 23682 . . . 4 (𝜑 → (𝐹𝑓 + 𝐺) ∈ dom ∫1)
4 i1frn 23664 . . . . . . . 8 (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
51, 4syl 17 . . . . . . 7 (𝜑 → ran 𝐹 ∈ Fin)
6 i1frn 23664 . . . . . . . 8 (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
72, 6syl 17 . . . . . . 7 (𝜑 → ran 𝐺 ∈ Fin)
8 xpfi 8387 . . . . . . 7 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
95, 7, 8syl2anc 573 . . . . . 6 (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
10 ax-addf 10217 . . . . . . . . . 10 + :(ℂ × ℂ)⟶ℂ
11 ffn 6185 . . . . . . . . . 10 ( + :(ℂ × ℂ)⟶ℂ → + Fn (ℂ × ℂ))
1210, 11ax-mp 5 . . . . . . . . 9 + Fn (ℂ × ℂ)
13 i1ff 23663 . . . . . . . . . . . . 13 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
141, 13syl 17 . . . . . . . . . . . 12 (𝜑𝐹:ℝ⟶ℝ)
15 frn 6193 . . . . . . . . . . . 12 (𝐹:ℝ⟶ℝ → ran 𝐹 ⊆ ℝ)
1614, 15syl 17 . . . . . . . . . . 11 (𝜑 → ran 𝐹 ⊆ ℝ)
17 ax-resscn 10195 . . . . . . . . . . 11 ℝ ⊆ ℂ
1816, 17syl6ss 3764 . . . . . . . . . 10 (𝜑 → ran 𝐹 ⊆ ℂ)
19 i1ff 23663 . . . . . . . . . . . . 13 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
202, 19syl 17 . . . . . . . . . . . 12 (𝜑𝐺:ℝ⟶ℝ)
21 frn 6193 . . . . . . . . . . . 12 (𝐺:ℝ⟶ℝ → ran 𝐺 ⊆ ℝ)
2220, 21syl 17 . . . . . . . . . . 11 (𝜑 → ran 𝐺 ⊆ ℝ)
2322, 17syl6ss 3764 . . . . . . . . . 10 (𝜑 → ran 𝐺 ⊆ ℂ)
24 xpss12 5264 . . . . . . . . . 10 ((ran 𝐹 ⊆ ℂ ∧ ran 𝐺 ⊆ ℂ) → (ran 𝐹 × ran 𝐺) ⊆ (ℂ × ℂ))
2518, 23, 24syl2anc 573 . . . . . . . . 9 (𝜑 → (ran 𝐹 × ran 𝐺) ⊆ (ℂ × ℂ))
26 fnssres 6144 . . . . . . . . 9 (( + Fn (ℂ × ℂ) ∧ (ran 𝐹 × ran 𝐺) ⊆ (ℂ × ℂ)) → ( + ↾ (ran 𝐹 × ran 𝐺)) Fn (ran 𝐹 × ran 𝐺))
2712, 25, 26sylancr 575 . . . . . . . 8 (𝜑 → ( + ↾ (ran 𝐹 × ran 𝐺)) Fn (ran 𝐹 × ran 𝐺))
28 itg1add.4 . . . . . . . . 9 𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))
2928fneq1i 6125 . . . . . . . 8 (𝑃 Fn (ran 𝐹 × ran 𝐺) ↔ ( + ↾ (ran 𝐹 × ran 𝐺)) Fn (ran 𝐹 × ran 𝐺))
3027, 29sylibr 224 . . . . . . 7 (𝜑𝑃 Fn (ran 𝐹 × ran 𝐺))
31 dffn4 6262 . . . . . . 7 (𝑃 Fn (ran 𝐹 × ran 𝐺) ↔ 𝑃:(ran 𝐹 × ran 𝐺)–onto→ran 𝑃)
3230, 31sylib 208 . . . . . 6 (𝜑𝑃:(ran 𝐹 × ran 𝐺)–onto→ran 𝑃)
33 fofi 8408 . . . . . 6 (((ran 𝐹 × ran 𝐺) ∈ Fin ∧ 𝑃:(ran 𝐹 × ran 𝐺)–onto→ran 𝑃) → ran 𝑃 ∈ Fin)
349, 32, 33syl2anc 573 . . . . 5 (𝜑 → ran 𝑃 ∈ Fin)
35 difss 3888 . . . . 5 (ran 𝑃 ∖ {0}) ⊆ ran 𝑃
36 ssfi 8336 . . . . 5 ((ran 𝑃 ∈ Fin ∧ (ran 𝑃 ∖ {0}) ⊆ ran 𝑃) → (ran 𝑃 ∖ {0}) ∈ Fin)
3734, 35, 36sylancl 574 . . . 4 (𝜑 → (ran 𝑃 ∖ {0}) ∈ Fin)
38 opelxpi 5288 . . . . . . . . . 10 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺) → ⟨𝑥, 𝑦⟩ ∈ (ran 𝐹 × ran 𝐺))
39 ffun 6188 . . . . . . . . . . . 12 ( + :(ℂ × ℂ)⟶ℂ → Fun + )
4010, 39ax-mp 5 . . . . . . . . . . 11 Fun +
4110fdmi 6192 . . . . . . . . . . . 12 dom + = (ℂ × ℂ)
4225, 41syl6sseqr 3801 . . . . . . . . . . 11 (𝜑 → (ran 𝐹 × ran 𝐺) ⊆ dom + )
43 funfvima2 6636 . . . . . . . . . . 11 ((Fun + ∧ (ran 𝐹 × ran 𝐺) ⊆ dom + ) → (⟨𝑥, 𝑦⟩ ∈ (ran 𝐹 × ran 𝐺) → ( + ‘⟨𝑥, 𝑦⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺))))
4440, 42, 43sylancr 575 . . . . . . . . . 10 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ (ran 𝐹 × ran 𝐺) → ( + ‘⟨𝑥, 𝑦⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺))))
4538, 44syl5 34 . . . . . . . . 9 (𝜑 → ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺) → ( + ‘⟨𝑥, 𝑦⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺))))
4645imp 393 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺)) → ( + ‘⟨𝑥, 𝑦⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺)))
47 df-ov 6796 . . . . . . . 8 (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
4828rneqi 5490 . . . . . . . . 9 ran 𝑃 = ran ( + ↾ (ran 𝐹 × ran 𝐺))
49 df-ima 5262 . . . . . . . . 9 ( + “ (ran 𝐹 × ran 𝐺)) = ran ( + ↾ (ran 𝐹 × ran 𝐺))
5048, 49eqtr4i 2796 . . . . . . . 8 ran 𝑃 = ( + “ (ran 𝐹 × ran 𝐺))
5146, 47, 503eltr4g 2867 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺)) → (𝑥 + 𝑦) ∈ ran 𝑃)
52 ffn 6185 . . . . . . . . 9 (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ)
5314, 52syl 17 . . . . . . . 8 (𝜑𝐹 Fn ℝ)
54 dffn3 6194 . . . . . . . 8 (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
5553, 54sylib 208 . . . . . . 7 (𝜑𝐹:ℝ⟶ran 𝐹)
56 ffn 6185 . . . . . . . . 9 (𝐺:ℝ⟶ℝ → 𝐺 Fn ℝ)
5720, 56syl 17 . . . . . . . 8 (𝜑𝐺 Fn ℝ)
58 dffn3 6194 . . . . . . . 8 (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
5957, 58sylib 208 . . . . . . 7 (𝜑𝐺:ℝ⟶ran 𝐺)
60 reex 10229 . . . . . . . 8 ℝ ∈ V
6160a1i 11 . . . . . . 7 (𝜑 → ℝ ∈ V)
62 inidm 3971 . . . . . . 7 (ℝ ∩ ℝ) = ℝ
6351, 55, 59, 61, 61, 62off 7059 . . . . . 6 (𝜑 → (𝐹𝑓 + 𝐺):ℝ⟶ran 𝑃)
64 frn 6193 . . . . . 6 ((𝐹𝑓 + 𝐺):ℝ⟶ran 𝑃 → ran (𝐹𝑓 + 𝐺) ⊆ ran 𝑃)
6563, 64syl 17 . . . . 5 (𝜑 → ran (𝐹𝑓 + 𝐺) ⊆ ran 𝑃)
6665ssdifd 3897 . . . 4 (𝜑 → (ran (𝐹𝑓 + 𝐺) ∖ {0}) ⊆ (ran 𝑃 ∖ {0}))
6716sselda 3752 . . . . . . . . . 10 ((𝜑𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
6822sselda 3752 . . . . . . . . . 10 ((𝜑𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
6967, 68anim12dan 605 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺)) → (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ))
70 readdcl 10221 . . . . . . . . 9 ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑦 + 𝑧) ∈ ℝ)
7169, 70syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺)) → (𝑦 + 𝑧) ∈ ℝ)
7271ralrimivva 3120 . . . . . . 7 (𝜑 → ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺(𝑦 + 𝑧) ∈ ℝ)
73 funimassov 6958 . . . . . . . 8 ((Fun + ∧ (ran 𝐹 × ran 𝐺) ⊆ dom + ) → (( + “ (ran 𝐹 × ran 𝐺)) ⊆ ℝ ↔ ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺(𝑦 + 𝑧) ∈ ℝ))
7440, 42, 73sylancr 575 . . . . . . 7 (𝜑 → (( + “ (ran 𝐹 × ran 𝐺)) ⊆ ℝ ↔ ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺(𝑦 + 𝑧) ∈ ℝ))
7572, 74mpbird 247 . . . . . 6 (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) ⊆ ℝ)
7650, 75syl5eqss 3798 . . . . 5 (𝜑 → ran 𝑃 ⊆ ℝ)
7776ssdifd 3897 . . . 4 (𝜑 → (ran 𝑃 ∖ {0}) ⊆ (ℝ ∖ {0}))
78 itg1val2 23671 . . . 4 (((𝐹𝑓 + 𝐺) ∈ dom ∫1 ∧ ((ran 𝑃 ∖ {0}) ∈ Fin ∧ (ran (𝐹𝑓 + 𝐺) ∖ {0}) ⊆ (ran 𝑃 ∖ {0}) ∧ (ran 𝑃 ∖ {0}) ⊆ (ℝ ∖ {0}))) → (∫1‘(𝐹𝑓 + 𝐺)) = Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · (vol‘((𝐹𝑓 + 𝐺) “ {𝑤}))))
793, 37, 66, 77, 78syl13anc 1478 . . 3 (𝜑 → (∫1‘(𝐹𝑓 + 𝐺)) = Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · (vol‘((𝐹𝑓 + 𝐺) “ {𝑤}))))
8020adantr 466 . . . . . . . 8 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → 𝐺:ℝ⟶ℝ)
817adantr 466 . . . . . . . 8 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → ran 𝐺 ∈ Fin)
82 inss2 3982 . . . . . . . . 9 ((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧})
8382a1i 11 . . . . . . . 8 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧}))
84 i1fima 23665 . . . . . . . . . . 11 (𝐹 ∈ dom ∫1 → (𝐹 “ {(𝑤𝑧)}) ∈ dom vol)
851, 84syl 17 . . . . . . . . . 10 (𝜑 → (𝐹 “ {(𝑤𝑧)}) ∈ dom vol)
86 i1fima 23665 . . . . . . . . . . 11 (𝐺 ∈ dom ∫1 → (𝐺 “ {𝑧}) ∈ dom vol)
872, 86syl 17 . . . . . . . . . 10 (𝜑 → (𝐺 “ {𝑧}) ∈ dom vol)
88 inmbl 23530 . . . . . . . . . 10 (((𝐹 “ {(𝑤𝑧)}) ∈ dom vol ∧ (𝐺 “ {𝑧}) ∈ dom vol) → ((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
8985, 87, 88syl2anc 573 . . . . . . . . 9 (𝜑 → ((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
9089ad2antrr 705 . . . . . . . 8 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
9135, 76syl5ss 3763 . . . . . . . . . . . . 13 (𝜑 → (ran 𝑃 ∖ {0}) ⊆ ℝ)
9291sselda 3752 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → 𝑤 ∈ ℝ)
9392adantr 466 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑤 ∈ ℝ)
9468adantlr 694 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
9593, 94resubcld 10660 . . . . . . . . . 10 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑤𝑧) ∈ ℝ)
9693recnd 10270 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑤 ∈ ℂ)
9794recnd 10270 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℂ)
9896, 97npcand 10598 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝑤𝑧) + 𝑧) = 𝑤)
99 eldifsni 4457 . . . . . . . . . . . . 13 (𝑤 ∈ (ran 𝑃 ∖ {0}) → 𝑤 ≠ 0)
10099ad2antlr 706 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑤 ≠ 0)
10198, 100eqnetrd 3010 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝑤𝑧) + 𝑧) ≠ 0)
102 oveq12 6802 . . . . . . . . . . . . 13 (((𝑤𝑧) = 0 ∧ 𝑧 = 0) → ((𝑤𝑧) + 𝑧) = (0 + 0))
103 00id 10413 . . . . . . . . . . . . 13 (0 + 0) = 0
104102, 103syl6eq 2821 . . . . . . . . . . . 12 (((𝑤𝑧) = 0 ∧ 𝑧 = 0) → ((𝑤𝑧) + 𝑧) = 0)
105104necon3ai 2968 . . . . . . . . . . 11 (((𝑤𝑧) + 𝑧) ≠ 0 → ¬ ((𝑤𝑧) = 0 ∧ 𝑧 = 0))
106101, 105syl 17 . . . . . . . . . 10 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ¬ ((𝑤𝑧) = 0 ∧ 𝑧 = 0))
107 itg1add.3 . . . . . . . . . . 11 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))
1081, 2, 107itg1addlem3 23685 . . . . . . . . . 10 ((((𝑤𝑧) ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ ¬ ((𝑤𝑧) = 0 ∧ 𝑧 = 0)) → ((𝑤𝑧)𝐼𝑧) = (vol‘((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))))
10995, 94, 106, 108syl21anc 1475 . . . . . . . . 9 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝑤𝑧)𝐼𝑧) = (vol‘((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))))
1101, 2, 107itg1addlem2 23684 . . . . . . . . . . 11 (𝜑𝐼:(ℝ × ℝ)⟶ℝ)
111110ad2antrr 705 . . . . . . . . . 10 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
112111, 95, 94fovrnd 6953 . . . . . . . . 9 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝑤𝑧)𝐼𝑧) ∈ ℝ)
113109, 112eqeltrrd 2851 . . . . . . . 8 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol‘((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
11480, 81, 83, 90, 113itg1addlem1 23679 . . . . . . 7 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → (vol‘ 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))) = Σ𝑧 ∈ ran 𝐺(vol‘((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))))
11592recnd 10270 . . . . . . . . 9 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → 𝑤 ∈ ℂ)
1161, 2i1faddlem 23680 . . . . . . . . 9 ((𝜑𝑤 ∈ ℂ) → ((𝐹𝑓 + 𝐺) “ {𝑤}) = 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})))
117115, 116syldan 579 . . . . . . . 8 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → ((𝐹𝑓 + 𝐺) “ {𝑤}) = 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})))
118117fveq2d 6336 . . . . . . 7 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → (vol‘((𝐹𝑓 + 𝐺) “ {𝑤})) = (vol‘ 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))))
119109sumeq2dv 14641 . . . . . . 7 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → Σ𝑧 ∈ ran 𝐺((𝑤𝑧)𝐼𝑧) = Σ𝑧 ∈ ran 𝐺(vol‘((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))))
120114, 118, 1193eqtr4d 2815 . . . . . 6 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → (vol‘((𝐹𝑓 + 𝐺) “ {𝑤})) = Σ𝑧 ∈ ran 𝐺((𝑤𝑧)𝐼𝑧))
121120oveq2d 6809 . . . . 5 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → (𝑤 · (vol‘((𝐹𝑓 + 𝐺) “ {𝑤}))) = (𝑤 · Σ𝑧 ∈ ran 𝐺((𝑤𝑧)𝐼𝑧)))
122112recnd 10270 . . . . . 6 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝑤𝑧)𝐼𝑧) ∈ ℂ)
12381, 115, 122fsummulc2 14723 . . . . 5 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → (𝑤 · Σ𝑧 ∈ ran 𝐺((𝑤𝑧)𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺(𝑤 · ((𝑤𝑧)𝐼𝑧)))
124121, 123eqtrd 2805 . . . 4 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → (𝑤 · (vol‘((𝐹𝑓 + 𝐺) “ {𝑤}))) = Σ𝑧 ∈ ran 𝐺(𝑤 · ((𝑤𝑧)𝐼𝑧)))
125124sumeq2dv 14641 . . 3 (𝜑 → Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · (vol‘((𝐹𝑓 + 𝐺) “ {𝑤}))) = Σ𝑤 ∈ (ran 𝑃 ∖ {0})Σ𝑧 ∈ ran 𝐺(𝑤 · ((𝑤𝑧)𝐼𝑧)))
12696, 122mulcld 10262 . . . . 5 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑤 · ((𝑤𝑧)𝐼𝑧)) ∈ ℂ)
127126anasss 457 . . . 4 ((𝜑 ∧ (𝑤 ∈ (ran 𝑃 ∖ {0}) ∧ 𝑧 ∈ ran 𝐺)) → (𝑤 · ((𝑤𝑧)𝐼𝑧)) ∈ ℂ)
12837, 7, 127fsumcom 14714 . . 3 (𝜑 → Σ𝑤 ∈ (ran 𝑃 ∖ {0})Σ𝑧 ∈ ran 𝐺(𝑤 · ((𝑤𝑧)𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · ((𝑤𝑧)𝐼𝑧)))
12979, 125, 1283eqtrd 2809 . 2 (𝜑 → (∫1‘(𝐹𝑓 + 𝐺)) = Σ𝑧 ∈ ran 𝐺Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · ((𝑤𝑧)𝐼𝑧)))
130 oveq1 6800 . . . . . . 7 (𝑦 = (𝑤𝑧) → (𝑦 + 𝑧) = ((𝑤𝑧) + 𝑧))
131 oveq1 6800 . . . . . . 7 (𝑦 = (𝑤𝑧) → (𝑦𝐼𝑧) = ((𝑤𝑧)𝐼𝑧))
132130, 131oveq12d 6811 . . . . . 6 (𝑦 = (𝑤𝑧) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = (((𝑤𝑧) + 𝑧) · ((𝑤𝑧)𝐼𝑧)))
13334adantr 466 . . . . . 6 ((𝜑𝑧 ∈ ran 𝐺) → ran 𝑃 ∈ Fin)
13476adantr 466 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ran 𝐺) → ran 𝑃 ⊆ ℝ)
135134sselda 3752 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) → 𝑣 ∈ ℝ)
13668adantr 466 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) → 𝑧 ∈ ℝ)
137135, 136resubcld 10660 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) → (𝑣𝑧) ∈ ℝ)
138137ex 397 . . . . . . . 8 ((𝜑𝑧 ∈ ran 𝐺) → (𝑣 ∈ ran 𝑃 → (𝑣𝑧) ∈ ℝ))
139135recnd 10270 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) → 𝑣 ∈ ℂ)
140139adantrr 696 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃𝑦 ∈ ran 𝑃)) → 𝑣 ∈ ℂ)
14176sselda 3752 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ran 𝑃) → 𝑦 ∈ ℝ)
142141ad2ant2rl 743 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃𝑦 ∈ ran 𝑃)) → 𝑦 ∈ ℝ)
143142recnd 10270 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃𝑦 ∈ ran 𝑃)) → 𝑦 ∈ ℂ)
14468recnd 10270 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℂ)
145144adantr 466 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃𝑦 ∈ ran 𝑃)) → 𝑧 ∈ ℂ)
146140, 143, 145subcan2ad 10639 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃𝑦 ∈ ran 𝑃)) → ((𝑣𝑧) = (𝑦𝑧) ↔ 𝑣 = 𝑦))
147146ex 397 . . . . . . . 8 ((𝜑𝑧 ∈ ran 𝐺) → ((𝑣 ∈ ran 𝑃𝑦 ∈ ran 𝑃) → ((𝑣𝑧) = (𝑦𝑧) ↔ 𝑣 = 𝑦)))
148138, 147dom2lem 8149 . . . . . . 7 ((𝜑𝑧 ∈ ran 𝐺) → (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃1-1→ℝ)
149 f1f1orn 6289 . . . . . . 7 ((𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃1-1→ℝ → (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃1-1-onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)))
150148, 149syl 17 . . . . . 6 ((𝜑𝑧 ∈ ran 𝐺) → (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃1-1-onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)))
151 oveq1 6800 . . . . . . . 8 (𝑣 = 𝑤 → (𝑣𝑧) = (𝑤𝑧))
152 eqid 2771 . . . . . . . 8 (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) = (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))
153 ovex 6823 . . . . . . . 8 (𝑤𝑧) ∈ V
154151, 152, 153fvmpt 6424 . . . . . . 7 (𝑤 ∈ ran 𝑃 → ((𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))‘𝑤) = (𝑤𝑧))
155154adantl 467 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → ((𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))‘𝑤) = (𝑤𝑧))
156 f1f 6241 . . . . . . . . . . 11 ((𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃1-1→ℝ → (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃⟶ℝ)
157 frn 6193 . . . . . . . . . . 11 ((𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃⟶ℝ → ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ⊆ ℝ)
158148, 156, 1573syl 18 . . . . . . . . . 10 ((𝜑𝑧 ∈ ran 𝐺) → ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ⊆ ℝ)
159158sselda 3752 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → 𝑦 ∈ ℝ)
16068adantr 466 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → 𝑧 ∈ ℝ)
161159, 160readdcld 10271 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → (𝑦 + 𝑧) ∈ ℝ)
162110ad2antrr 705 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → 𝐼:(ℝ × ℝ)⟶ℝ)
163162, 159, 160fovrnd 6953 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → (𝑦𝐼𝑧) ∈ ℝ)
164161, 163remulcld 10272 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) ∈ ℝ)
165164recnd 10270 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) ∈ ℂ)
166132, 133, 150, 155, 165fsumf1o 14662 . . . . 5 ((𝜑𝑧 ∈ ran 𝐺) → Σ𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑤 ∈ ran 𝑃(((𝑤𝑧) + 𝑧) · ((𝑤𝑧)𝐼𝑧)))
167134sselda 3752 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → 𝑤 ∈ ℝ)
168167recnd 10270 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → 𝑤 ∈ ℂ)
169144adantr 466 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → 𝑧 ∈ ℂ)
170168, 169npcand 10598 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → ((𝑤𝑧) + 𝑧) = 𝑤)
171170oveq1d 6808 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → (((𝑤𝑧) + 𝑧) · ((𝑤𝑧)𝐼𝑧)) = (𝑤 · ((𝑤𝑧)𝐼𝑧)))
172171sumeq2dv 14641 . . . . 5 ((𝜑𝑧 ∈ ran 𝐺) → Σ𝑤 ∈ ran 𝑃(((𝑤𝑧) + 𝑧) · ((𝑤𝑧)𝐼𝑧)) = Σ𝑤 ∈ ran 𝑃(𝑤 · ((𝑤𝑧)𝐼𝑧)))
173166, 172eqtrd 2805 . . . 4 ((𝜑𝑧 ∈ ran 𝐺) → Σ𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑤 ∈ ran 𝑃(𝑤 · ((𝑤𝑧)𝐼𝑧)))
17442ad2antrr 705 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (ran 𝐹 × ran 𝐺) ⊆ dom + )
175 simpr 471 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ran 𝐹)
176 simplr 752 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ran 𝐺)
177 opelxpi 5288 . . . . . . . . . . . 12 ((𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺) → ⟨𝑦, 𝑧⟩ ∈ (ran 𝐹 × ran 𝐺))
178175, 176, 177syl2anc 573 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ⟨𝑦, 𝑧⟩ ∈ (ran 𝐹 × ran 𝐺))
179 funfvima2 6636 . . . . . . . . . . . 12 ((Fun + ∧ (ran 𝐹 × ran 𝐺) ⊆ dom + ) → (⟨𝑦, 𝑧⟩ ∈ (ran 𝐹 × ran 𝐺) → ( + ‘⟨𝑦, 𝑧⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺))))
18040, 179mpan 670 . . . . . . . . . . 11 ((ran 𝐹 × ran 𝐺) ⊆ dom + → (⟨𝑦, 𝑧⟩ ∈ (ran 𝐹 × ran 𝐺) → ( + ‘⟨𝑦, 𝑧⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺))))
181174, 178, 180sylc 65 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ( + ‘⟨𝑦, 𝑧⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺)))
182 df-ov 6796 . . . . . . . . . 10 (𝑦 + 𝑧) = ( + ‘⟨𝑦, 𝑧⟩)
183181, 182, 503eltr4g 2867 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦 + 𝑧) ∈ ran 𝑃)
18467adantlr 694 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
185184recnd 10270 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℂ)
186144adantr 466 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
187185, 186pncand 10595 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦 + 𝑧) − 𝑧) = 𝑦)
188187eqcomd 2777 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 = ((𝑦 + 𝑧) − 𝑧))
189 oveq1 6800 . . . . . . . . . . 11 (𝑣 = (𝑦 + 𝑧) → (𝑣𝑧) = ((𝑦 + 𝑧) − 𝑧))
190189eqeq2d 2781 . . . . . . . . . 10 (𝑣 = (𝑦 + 𝑧) → (𝑦 = (𝑣𝑧) ↔ 𝑦 = ((𝑦 + 𝑧) − 𝑧)))
191190rspcev 3460 . . . . . . . . 9 (((𝑦 + 𝑧) ∈ ran 𝑃𝑦 = ((𝑦 + 𝑧) − 𝑧)) → ∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧))
192183, 188, 191syl2anc 573 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧))
193192ralrimiva 3115 . . . . . . 7 ((𝜑𝑧 ∈ ran 𝐺) → ∀𝑦 ∈ ran 𝐹𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧))
194 ssabral 3822 . . . . . . 7 (ran 𝐹 ⊆ {𝑦 ∣ ∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧)} ↔ ∀𝑦 ∈ ran 𝐹𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧))
195193, 194sylibr 224 . . . . . 6 ((𝜑𝑧 ∈ ran 𝐺) → ran 𝐹 ⊆ {𝑦 ∣ ∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧)})
196152rnmpt 5509 . . . . . 6 ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) = {𝑦 ∣ ∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧)}
197195, 196syl6sseqr 3801 . . . . 5 ((𝜑𝑧 ∈ ran 𝐺) → ran 𝐹 ⊆ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)))
19868adantr 466 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
199184, 198readdcld 10271 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦 + 𝑧) ∈ ℝ)
200110ad2antrr 705 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
201200, 184, 198fovrnd 6953 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
202199, 201remulcld 10272 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) ∈ ℝ)
203202recnd 10270 . . . . 5 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) ∈ ℂ)
204158ssdifd 3897 . . . . . . 7 ((𝜑𝑧 ∈ ran 𝐺) → (ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ∖ ran 𝐹) ⊆ (ℝ ∖ ran 𝐹))
205204sselda 3752 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ∖ ran 𝐹)) → 𝑦 ∈ (ℝ ∖ ran 𝐹))
206 eldifi 3883 . . . . . . . . . . . . 13 (𝑦 ∈ (ℝ ∖ ran 𝐹) → 𝑦 ∈ ℝ)
207206ad2antrl 707 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → 𝑦 ∈ ℝ)
20868adantr 466 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → 𝑧 ∈ ℝ)
209 simprr 756 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
2101, 2, 107itg1addlem3 23685 . . . . . . . . . . . 12 (((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0)) → (𝑦𝐼𝑧) = (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
211207, 208, 209, 210syl21anc 1475 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝑦𝐼𝑧) = (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
212 inss1 3981 . . . . . . . . . . . . . . 15 ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {𝑦})
213 eldifn 3884 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (ℝ ∖ ran 𝐹) → ¬ 𝑦 ∈ ran 𝐹)
214213ad2antrl 707 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ¬ 𝑦 ∈ ran 𝐹)
215 vex 3354 . . . . . . . . . . . . . . . . . . . 20 𝑦 ∈ V
216 vex 3354 . . . . . . . . . . . . . . . . . . . . 21 𝑣 ∈ V
217216eliniseg 5635 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ V → (𝑣 ∈ (𝐹 “ {𝑦}) ↔ 𝑣𝐹𝑦))
218215, 217ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑣 ∈ (𝐹 “ {𝑦}) ↔ 𝑣𝐹𝑦)
219216, 215brelrn 5494 . . . . . . . . . . . . . . . . . . 19 (𝑣𝐹𝑦𝑦 ∈ ran 𝐹)
220218, 219sylbi 207 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ (𝐹 “ {𝑦}) → 𝑦 ∈ ran 𝐹)
221214, 220nsyl 137 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ¬ 𝑣 ∈ (𝐹 “ {𝑦}))
222221pm2.21d 119 . . . . . . . . . . . . . . . 16 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝑣 ∈ (𝐹 “ {𝑦}) → 𝑣 ∈ ∅))
223222ssrdv 3758 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝐹 “ {𝑦}) ⊆ ∅)
224212, 223syl5ss 3763 . . . . . . . . . . . . . 14 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ ∅)
225 ss0 4118 . . . . . . . . . . . . . 14 (((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ ∅ → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ∅)
226224, 225syl 17 . . . . . . . . . . . . 13 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ∅)
227226fveq2d 6336 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) = (vol‘∅))
228 0mbl 23527 . . . . . . . . . . . . . 14 ∅ ∈ dom vol
229 mblvol 23518 . . . . . . . . . . . . . 14 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
230228, 229ax-mp 5 . . . . . . . . . . . . 13 (vol‘∅) = (vol*‘∅)
231 ovol0 23481 . . . . . . . . . . . . 13 (vol*‘∅) = 0
232230, 231eqtri 2793 . . . . . . . . . . . 12 (vol‘∅) = 0
233227, 232syl6eq 2821 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) = 0)
234211, 233eqtrd 2805 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝑦𝐼𝑧) = 0)
235234oveq2d 6809 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = ((𝑦 + 𝑧) · 0))
236207, 208readdcld 10271 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝑦 + 𝑧) ∈ ℝ)
237236recnd 10270 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝑦 + 𝑧) ∈ ℂ)
238237mul01d 10437 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ((𝑦 + 𝑧) · 0) = 0)
239235, 238eqtrd 2805 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = 0)
240239expr 444 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ℝ ∖ ran 𝐹)) → (¬ (𝑦 = 0 ∧ 𝑧 = 0) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = 0))
241 oveq12 6802 . . . . . . . . . 10 ((𝑦 = 0 ∧ 𝑧 = 0) → (𝑦 + 𝑧) = (0 + 0))
242241, 103syl6eq 2821 . . . . . . . . 9 ((𝑦 = 0 ∧ 𝑧 = 0) → (𝑦 + 𝑧) = 0)
243 oveq12 6802 . . . . . . . . . 10 ((𝑦 = 0 ∧ 𝑧 = 0) → (𝑦𝐼𝑧) = (0𝐼0))
244 0re 10242 . . . . . . . . . . 11 0 ∈ ℝ
245 iftrue 4231 . . . . . . . . . . . 12 ((𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) = 0)
246 c0ex 10236 . . . . . . . . . . . 12 0 ∈ V
247245, 107, 246ovmpt2a 6938 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ 0 ∈ ℝ) → (0𝐼0) = 0)
248244, 244, 247mp2an 672 . . . . . . . . . 10 (0𝐼0) = 0
249243, 248syl6eq 2821 . . . . . . . . 9 ((𝑦 = 0 ∧ 𝑧 = 0) → (𝑦𝐼𝑧) = 0)
250242, 249oveq12d 6811 . . . . . . . 8 ((𝑦 = 0 ∧ 𝑧 = 0) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = (0 · 0))
251 0cn 10234 . . . . . . . . 9 0 ∈ ℂ
252251mul01i 10428 . . . . . . . 8 (0 · 0) = 0
253250, 252syl6eq 2821 . . . . . . 7 ((𝑦 = 0 ∧ 𝑧 = 0) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = 0)
254240, 253pm2.61d2 173 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ℝ ∖ ran 𝐹)) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = 0)
255205, 254syldan 579 . . . . 5 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ∖ ran 𝐹)) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = 0)
256 f1ofo 6285 . . . . . . 7 ((𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃1-1-onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) → (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)))
257150, 256syl 17 . . . . . 6 ((𝜑𝑧 ∈ ran 𝐺) → (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)))
258 fofi 8408 . . . . . 6 ((ran 𝑃 ∈ Fin ∧ (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ∈ Fin)
259133, 257, 258syl2anc 573 . . . . 5 ((𝜑𝑧 ∈ ran 𝐺) → ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ∈ Fin)
260197, 203, 255, 259fsumss 14664 . . . 4 ((𝜑𝑧 ∈ ran 𝐺) → Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
26135a1i 11 . . . . 5 ((𝜑𝑧 ∈ ran 𝐺) → (ran 𝑃 ∖ {0}) ⊆ ran 𝑃)
262126an32s 631 . . . . 5 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) → (𝑤 · ((𝑤𝑧)𝐼𝑧)) ∈ ℂ)
263 dfin4 4016 . . . . . . . 8 (ran 𝑃 ∩ {0}) = (ran 𝑃 ∖ (ran 𝑃 ∖ {0}))
264 inss2 3982 . . . . . . . 8 (ran 𝑃 ∩ {0}) ⊆ {0}
265263, 264eqsstr3i 3785 . . . . . . 7 (ran 𝑃 ∖ (ran 𝑃 ∖ {0})) ⊆ {0}
266265sseli 3748 . . . . . 6 (𝑤 ∈ (ran 𝑃 ∖ (ran 𝑃 ∖ {0})) → 𝑤 ∈ {0})
267 elsni 4333 . . . . . . . . 9 (𝑤 ∈ {0} → 𝑤 = 0)
268267adantl 467 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → 𝑤 = 0)
269268oveq1d 6808 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → (𝑤 · ((𝑤𝑧)𝐼𝑧)) = (0 · ((𝑤𝑧)𝐼𝑧)))
270110ad2antrr 705 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → 𝐼:(ℝ × ℝ)⟶ℝ)
271268, 244syl6eqel 2858 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → 𝑤 ∈ ℝ)
27268adantr 466 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → 𝑧 ∈ ℝ)
273271, 272resubcld 10660 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → (𝑤𝑧) ∈ ℝ)
274270, 273, 272fovrnd 6953 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → ((𝑤𝑧)𝐼𝑧) ∈ ℝ)
275274recnd 10270 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → ((𝑤𝑧)𝐼𝑧) ∈ ℂ)
276275mul02d 10436 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → (0 · ((𝑤𝑧)𝐼𝑧)) = 0)
277269, 276eqtrd 2805 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → (𝑤 · ((𝑤𝑧)𝐼𝑧)) = 0)
278266, 277sylan2 580 . . . . 5 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ (ran 𝑃 ∖ (ran 𝑃 ∖ {0}))) → (𝑤 · ((𝑤𝑧)𝐼𝑧)) = 0)
279261, 262, 278, 133fsumss 14664 . . . 4 ((𝜑𝑧 ∈ ran 𝐺) → Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · ((𝑤𝑧)𝐼𝑧)) = Σ𝑤 ∈ ran 𝑃(𝑤 · ((𝑤𝑧)𝐼𝑧)))
280173, 260, 2793eqtr4d 2815 . . 3 ((𝜑𝑧 ∈ ran 𝐺) → Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · ((𝑤𝑧)𝐼𝑧)))
281280sumeq2dv 14641 . 2 (𝜑 → Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · ((𝑤𝑧)𝐼𝑧)))
282203anasss 457 . . 3 ((𝜑 ∧ (𝑧 ∈ ran 𝐺𝑦 ∈ ran 𝐹)) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) ∈ ℂ)
2837, 5, 282fsumcom 14714 . 2 (𝜑 → Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
284129, 281, 2833eqtr2d 2811 1 (𝜑 → (∫1‘(𝐹𝑓 + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  {cab 2757  wne 2943  wral 3061  wrex 3062  Vcvv 3351  cdif 3720  cin 3722  wss 3723  c0 4063  ifcif 4225  {csn 4316  cop 4322   ciun 4654   class class class wbr 4786  cmpt 4863   × cxp 5247  ccnv 5248  dom cdm 5249  ran crn 5250  cres 5251  cima 5252  Fun wfun 6025   Fn wfn 6026  wf 6027  1-1wf1 6028  ontowfo 6029  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793  cmpt2 6795  𝑓 cof 7042  Fincfn 8109  cc 10136  cr 10137  0cc0 10138   + caddc 10141   · cmul 10143  cmin 10468  Σcsu 14624  vol*covol 23450  volcvol 23451  1citg1 23603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216  ax-addf 10217
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-disj 4755  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8504  df-inf 8505  df-oi 8571  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-z 11580  df-uz 11889  df-q 11992  df-rp 12036  df-xadd 12152  df-ioo 12384  df-ico 12386  df-icc 12387  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-sum 14625  df-xmet 19954  df-met 19955  df-ovol 23452  df-vol 23453  df-mbf 23607  df-itg1 23608
This theorem is referenced by:  itg1addlem5  23687
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